Question

Consider the setsAandB,whereA={3,|B|}andB={1,|A|,|B|}.Whatare the sets?

Answer 1

It follows that $|A|=\begin{cases}2,\ |B|\ne 3\\ 1,\ |B|=3\end{cases}.$ Suppose that $|A|=1,$ then $B=\{1,1,|B|\}=\{1,|B|\},$ and we have a contradiction with $|B|=3.$ Therefore, $|A|=2,$ and thus $|B|\ne 3.$ Taking into account that $B=\{1,|A|,|B|\}=\{1,2,|B|\}$ and $|B|<3,$ we conclude that $|B|=2.$

Consequently, $A=\{3,2\}$ and $B=\{1,2\}.$